High School Mathematics-Physics SMILE Meeting
09 December 2003
Notes Prepared by Porter Johnson

Information:

Earl Zwicker obtained an email
message from a former SMILE participant and colleague, Arnold
Burkert, calling attention to a presentation of how a
mathematician, engineer, or scientist might write the equation, 1
+ 1 = 2, in a more "clear, scientific form". The result,
then, should be more comprehensible to all concerned, shouldn't it? You
may view a Microsoft Power-Point® presentation that appears
on the [bilingual Afrikaans-English] website of the Department
of Electrical, Electronic, and Computer Engineering of the University
of Pretoria, South Africa.
It may take a while to find and load this one, but it is well worth
watching!
Check it out! Thanks, Arnold and Earl!

Deborah Koolbeck [Chicago Academy of Arts HS,
physics] discussed plans for assembling resources
and services targeted specifically for high school mathematics and
science teachers who are unable to obtain basic instructional support
--- especially teachers in the Chicago Public School
system. These programs would be sponsored by the newly created Outreach
Program Committee at Fermilab, under the guidance of its head, Dr.Chris White. [Note:Chris is a
full-time Physics faculty member at IIT, winner of the 2002
IIT Excellence in Teaching Award, a superb phenomenological physics
teacher, and a true friend and strong supporter of the SMILE program]!
The first step of this outreach program is to make an inventory of the
most urgent needs of teachers, and to determine what resources can be
obtained to meet some of these needs. Specifically, Deb K
[wearing her "other hat" of outreach coordinator at Fermilab]
passed around a preliminary questionnaire to survey certain
specific needs of High School SMILE participants. Fermilab
has been able to obtain limited, precious resources that are
specifically earmarked for Outreach. They want to have the
greatest impact in providing significant, timely assistance to address
the most urgent needs of beleaguered, stalwart mathematics/science
teachers in local schools. By filling out and commenting upon a
preliminary questionnaire, SMILE participants are serving as a
vital focus group to improve the questionnaire itself, as well as to
focus it upon the needs of the schools and the possibilities of
developing an outreach program to address them. Here are some of
the issues in which they might be able to help you:

Would you like one of our technical experts to visit your
school to talk to students about (1) accelerators, particles, and
cosmology; (2) the daily life of our scientists/engineers; or (3)
role of computers in basic research?

Can we come to assist you in (1) setting up demonstration
equipment or computers, or (2) presenting basic ideas and
concepts of physics to your students?

Would you like to have a direct email connection to an
appropriate expert, so that you and/or your students could ask "far
reaching" questions and obtain assistance?

Do you already make use of Fermilab for educational
resources? If so, how?

How can we help strengthen science education at your
school? How can we help you professionally, as well as in your
science instruction?

Thanks, Deb! You may also contact Deborah Koolbeck
directly at her Email address: koolbeck@fnal.gov.
We look forward to working with you and Chris on these very
exciting developments!

Michelle Jones [Morgan Park HS], a long-term
participant and former staff member of SMILE whom we dearly
remember, visited us to call attention to New Leaders for New
Schools:TYPE 75 PRINCIPAL PROGRAM that addresses the
challenge of urban school leadership. More information, as well
as an Online Application, is located on their website: http://www.nlns.org/. In
addition, you may call the Chicago Office at 1 - 312 - 829-6547, or you
may Email Michelle directly at mjones@nlns.org,
Thanks for telling us about this program. We still love you,
Michelle, and hope that you have a "cross-over" hit!

Porter Johnson asked the (not entirely hypothetical)
question of whether a large (Volvo®) sedan could become
"totaled" while running into the rear of a compact (Toyota®)
sedan. Interestingly enough, this can happen, since the
accelerometer sensor to release the air bags is located behind the
grill on some Volvo sedans, as well as on certain other
cars. If that grill is even slightly damaged, the sensor will
cause both air bags to be deployed [essentially by firing
explosive charges that resemble shotgun shells] --- which, in turn,
often break the front windshield. When this occurs with a
relatively mature vehicle, the cost of repair becomes greater than the
"replacement cost" of the vehicle less the "salvage value" of the
vehicle "cannibalized" for spare parts. The vehicle is thus
deemed a total loss by insurance adjustors. This occurs, in spite
of our naive assumptions based upon Newton's Laws of Motion.
Fairy tales can come true; it can happen to you ...
occasionally!

Don Kanner (Lane Tech HS Physics
Teacher) Helicopter Whirligigs, General
Relativity, and
Forces
At our last meeting John Scavo was wondering how the number of
blades on a propeller related to airflow and
Richard
Goberville had propeller whirligigs for all of us to take and
experiment with.
Don had collected all of the
whirligigs from the other Lane Tech teachers at SMILE
and modified them in an attempt to answer John's question.
Here is the data
table:

Helicopter Whirligig Data

Number of Blades

Mass / grams

Result

4 blades

11.3

hovers

2 blades

11.3

hits ceiling

2 blades

5.8

hits ceiling

2 blades

4.0

hits ceiling

2 half-blades

9.5

halfway up

1 blade (balanced)

9.5

hovers

It was shown that while a four blade whirligig would only hover,
the two blade reached the ceiling. With the speculation that the mass
of the
whirligig was the factor, a second two blade was prepared by adding
mass to match with the four blade. Surprisingly, it also reached the
ceiling. Thus, we have good reason to believe that
two blades move more air than four blades. However, rotational speed
was not controlled.
Christopher Jarr, a student at Lane Tech, suggested
that the whirligigs might be slipped over the shaft of a drill bit on
an electric drill to produce a
constant speed for each. Lane Tech Earth science teachers
pointed out that hand-held anemometers were available in most
of the science supply house catalogs.

Having just used Karlene Joseph's and Dan Caldwell's
paper plate and marble centripetal force demonstration in his
classroom, Don showed us how to illustrate a celestial object
being pulled into a black hole (sparing no expense -- ha!)
using a marble whirling
around inside a the top of a 1 liter plastic pop bottle held
vertically with its mouth pointed toward the floor. When one stops
rotating the bottle, the marble continues to whirl around the inside
until it
falls out the mouth. In
Einstein's General Theory of Relativity, gravity corresponds to
a distortion
(intrinsic curvature) of the space around a mass. We are thus
led to
following the question: Assuming that the mouth of the bottle is
analogous
to a black hole, what portion of the area near the mouth of the
bottle best fits Einstein's description of space-time distortion?

Now who says that you can't teach about
black holes in high school? Don referred to the classic film Frames of Reference by
Hume and Ivey. The following description is adapted from
information
on the website of the Department of Physics and Astronomy
of the University
of Victoria (BC, Canada), [http://www.phys.uvic.ca]:

FRAMES OF REFERENCE (Educational Services, Inc.,
1960) 25 min, snd, b.w.Professors Patterson Hume and Donald Ivey of the
University of Toronto demonstrate the behavior of a body under the
force of gravity as viewed from different frames of reference and the
behavior of a frictionless puck on a rotating table in the laboratory.
Two excerpts from this film are also available which present the above
material in a condensed form:
1. "Excerpt 1", 7 min. QA839 F7. Shows gravitational effects.
2. "Excerpt 2", 5 1/2 min. QA839 F72. Shows rotational effects.

PJ Comment: The idea of "tunneling into the center of
force" is
very old. Isaac Newton criticized the Descartes model
of
the solar system, pointing out that this would be precisely the outcome
of that
mode. The Bohr Atomic Model was roundly criticized
during the period
1915-1925,
because electrons in atomic orbits would be expected to "wind down"
into the
center of the nucleus, in a time of about 10-9 seconds,
because of the
total power radiated by a free point charge q experiencing an
acceleration of magnitude a. According to the Larmor
formula

P = (2 k q2 a2 ) / (3 c3)

Note that the velocity of light c ~ 3 x 108 m/sec,
and k ~ 9 x
109 Nt m2/Coul2 is the constant
appearing in Coulomb's Law. In addition, it is a consequence
of Quantum
Mechanics that magnetic monopoles, if they happen to
exist, will -- in
effect -- gobble up both negatively and
positively
charged particles that come directly at them (zero angular momentum).
This idea of
orbiting
into the center of attraction is a timeless, recurrent theme in
physics! For
additional information see the History of Mathematics Website
at St Andrews
University [
http://www-groups.dcs.st-and.ac.uk/~history/]
and especially the entry for
Sir
Isaac Newton, as well as
Theories
of Gravitation.

Finally, Don had a modified version of Gary
Guzdziol's vacuum
disk [mp111803.html]. By reducing the
internal plastic disk's diameter,
it was used as a washer for a nut on the end of an eyebolt to which a
heavy cord was attached. Without the worry of
string breaking, Don appeared to lift a stool with the device
but left us wondering
which force really lifted the
stool.

Don also posed the challenge of how to get all the marbles
to stay on a
paper plate when
the plate is rotated, as an extension of the lesson given by
Karlene Joseph at the last
SMILE meeting [mp111803.html]. Go
for it, Don --- you're on a roll!

Leticia Rodriguez [Peck Elementary
School] Chemical TestsLeticia brought in a tray of materials which we were to
categorize by their physical
properties of form, color and texture. She had put various types
of food
coloring close to these samples, so that SMILE participants
(role-playing as third
graders) could more easily distinguish them. We were permitted to
see,
feel, hear, and smell them, after being assured that these
particular materials passed
relevant safety tests. Here is a table summarizing our
observations:

Observations of Properties of Unknown
Solids

Color

Description:

(secret identity)

Red

crystals, crunchy,
rocky, clean, clear

sugar

Yellow

transparent ,solid,
dull, white, powder

alum

Green

solid, white, smooth,
dull, perfume scent

talc

Blue

crystal, solid, white,
cracking, no odor

baking soda

Orange

powder, no odor

corn starch

This lesson is adapted from one described on the Carolina
Scientific Website
[http://www.carolina.com/] on
the STC Units Descriptions.
Good stuff, Leticia!

Fred Schaal [Lane Tech HS,
mathematics] Unleashing
Complex Numbers
Fred extended the consideration of zeroes of quadratic
functions; ax2 + bx + c = 0 , which he began at
the last SMILE
class. He wrote down the quadratic formula

x±
= [ -b ± Ö(b2 -4ac)
] /(2a)

and
asked what happens in the case (a, b, c) = (1, 2, 3)?, In that
case one
obtains x = -1 ± Ö(-2).
This case, as
well as many, many others, involves taking the square root of a
negative
number. By adopting the notation Ö(-1)
= i or i2 = -1, he introduced complex
numbers and
wrote the answer as x = -1 ± 2i.
He
then showed, using the algebra of complex numbers, that these two
complex
numbers satisfy the original quadratic formula:

(-1 ± 2i) 2
+ 2 (-1 ± 2i)
+ 3 ?=? 0

1 - (± 4i ) -4 + 2 (± 2i) + 3 ?=? 0

0 = 0

All right!So, complex numbers are not so complex, after
all! Thanks, Fred!

Porter Johnson mentioned that complex
numbers were originally used merely to solve polynomial equations,
after Gauss
showed that every n-th order polynomial equation has n
(possibly
degenerate) complex roots. Much later, a mechanical engineer
named Fourier
made explicit use of the Euler formula, eix =
cos x + i sin
x, to develop Fourier series for the specific purpose
of
solving problems related to time-dependent heat flow in
conductors. The
electrical engineers introduced the complex impedance of a
circuit as a
means of analysis of time-dependent circuit behavior. In addition,
complex
numbers play a special role in descriptions of electromagnetic waves
through Maxwell's
Equations of electromagnetism. In 1925, the young
physicists Schrödinger and Heisenberg
independently
developed Quantum Mechanics. For the first time, complex
numbers
played a central and unavoidable role in that theory, and in
virtually all
subsequent theoretical developments in physics. In effect, the central
theoretical concept (wave function, probability amplitude, state of
the
system) cannot be measured directly, although its effects can be seen all
over the universe! For additional information see these
St Andrews
University History of Mathematics pages: Quadratic,
cubic, and Quartic Equations and The
Fundamental Theorem of Algebra.

Arlyn VanEk [Illiana Christian HS,
physics] Air Resistance of
Swinging Block
Arlyn VanEk set up a bifilar pendulum in front of us. He explained
that he had done this in his classroom. He used a wooden block (with
two eye screws) on its top edge for its bob, and suspended it from the
ceiling using light, inelastic cord tied to the eyes screws. He then
swung the bob back in an arc, keeping the cord taut to make angle to
the vertical. Then he released the bob from rest, and measured the time
for it to pass through a Pasco® (http://www.pasco.com) timing gate at
the bottom of its swing. Knowing the width of the block, its speed
could be calculated. From this kinetic energy could be calculated at
the bottom of its swing, and its gravitational potential energy could
be calculated by measuring the decrease in altitude from the beginning
to the bottom of its swing.

Assuming air friction is nil and noting that the motion of the block
should not depend upon its mass, when he did careful measurements, he
found that energy was not being conserved! Thinking that this might be
due to air friction, Arlyn made a second block with a
streamlined shape, resembling the head of a doubled-bladed axe.
Repeating the experiment with this bob, he found this time that it had
more energy at the bottom of its swing than it had potential energy at
the beginning! How could this be?!

Ann Brandon and Larry Alofs remarked that the separation of
kinetic
energy of a moving body into rotational
and translational motion can be made only corresponding to a
translation of the
center of mass, and rotation with respect to the center of mass.
One may thus apply the principle of conservation of energy using that
principle. Correspondingly, it is crucial to be certain that the
center of
mass of the pendulum moves through a circular arc, and that
translational speeds
are measured for the motion of that center mass. The standard
arrangement
for a ballistic pendulum is to suspend a wooden block by four
strings, attached on the top side on
locations symmetric with respect to the front, back, left, and right
edges. One thereby makes a bifilar pendulum, with the desired
properties. PJ
also mentioned that an aerodynamic shape should more properly
represent an
aircraft wing --- sharp in the front and smooth in the back, rather
than being
front-back symmetric. He pointed out that the winning athletes in
the
platform ski jumps in the 2002 Winter Olympics (in addition to
being
anorexic) held their skis- cross-pointed at the front, to reduce
air
resistance, rather than in the standard
railroad track position, to reduce air resistance.

As a sequel, Arlyn
showed that two steel balls collide almost elastically when one rolls
into the other at
rest on a smooth table. However, if the balls smash against one
another in
mid-flight and stay together, all the mechanical energy must be
converted into
heat. How do we know this?Arlyn took two
solid steel balls
[mass of about 500 grams each; about 5 cm in
diameter], and smashed them together
while a sheet of ordinary paper was held between them. It
was quite
plain to see that a small hole had been burned through the paper with
each
encounter. Furthermore, when the experiment was repeated in a
darkened
room, we could see flashes of light with each collision. Also,
the smell
of burnt paper was unmistakable Remarkable!

Finally, on a
non-destructive note, Arlyn held up a Thumb Drive Flash
Memory Stick
with a capacity of 64 MB that can be inserted into the USB plug
on
his fairly new computer. He is using this small memory
stick (normally
used for a digital camera) to transfer data from the school
computer to his computer at home -- the hard drive recognizes it as a
formatted [ROM] disc, so that files can be moved to and fro.
Neato!

Arlyn, you showed us how it really is! Thanks!

Fred Farnell [Lane Tech HS,
physics] Rocket Balloons
Fred took a long,
collapsed balloon, and inflated it by inserting a special straw and
blowing.
Then he released it into the
air. It zoomed around the room, making a "screaming"
sound.
Just for amusement and edification, he sent off several more balloons,
with
similar effect --- except for the one that exploded during
inflation. This
is an ideal party favor, which Fred had obtained from The
Party
Corner®, in
Orland Park Shopping Center. It was described on the package
as follows:

Referring to his presentation at a previous SMILE meeting [mp111803.html],
Fred promised that he would bring his daughter's old tennis
shoes to SMILE
in the
near future, since she is nearly ready to donate them to us for
scientific study.
And, it's about time for her to wear winter shoes!

Those rockets really took off! Thanks
for showing us, Fred.

Bill Shanks [retired physics teacher & member,
Joliet Junior Chorale]
Clothes Pins
that Light Up
Bill showed us the perfect "party gag" gift ---
a pack of 50 plastic clothes pins, complete with (LED) bulbs,
which light
when you use the clothes pin to clamp the LED leads to make
contact with
electrodes on a small "dime shaped" Lithium cell, such as CL 2016,
2025, or 2032, which are rated at 2.8 Volts.
PJ
Comment: A red or
green LED can be lighted with a single cell, since a photon of
energy 2.8
eV corresponds to a wavelength

l = c / n = h c /(h n)
= hc / E = 1240 eV nm / 2.8 eV = 442 nm.

How could you use this in class?

Bill, you must be the life of the party! Very nice!

Karlene Joseph [Lane Tech HS,
physics] Crash and Burn
Website
Karlene showed us some video images of collisions of automobiles
in
which "crash dummies", as well as stunt drivers were sitting in
the
automobiles. The frame-by-frame sequence of images is quite
fascinating. It was
clear to all of us that, in fact, Newton's Laws fully explain
the
occurrences during the crash. In particular, when we saw the
impact and
damage when the head of the unrestrained occupant hit the dashboard,
the warning "wear
your seatbelts" was justified in graphic detail. These images
were
located on the website of The Center for Injury Control,School
of Public Health, Emory University [http://www.sph.emory.edu/]
on the Motor Vehicle Crash Video page.

Those daredevils and dummies showed how Newton's laws determine
the course
of collisions! Thanks,
Karlene!